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STATE SPACE DESCRIPTION OF A DYNAMIC SYSTEM
1. Linear case
x& (t )
u(t) x(t) y(t)
B ∫
x (t ) = x& (t ) dt C
x& (t ) = A x(t) + B u(t)
y(t) = C x(t) (in some cases y(t) = C x(t) + D u(t) → improper system)
x = vector of n state space variables; x& = vector of n state derivatives;
A = n x n matrix of linear state equation (sometimes F);
u = vector of m input variables; B = n x m input matrix (sometimes G);
y = vector of p output variables; C = p x n input matrix (sometimes H).
2. Single input single output linear case (Linear SISO systems)
x& (t ) = A x(t) + b u(t)
y(t) = c’ x(t)
m = 1, u scalar, b = B column vector; p = 1, y scalar, c’ = C row vector
Transfer function: H(s) = c’ (sI - A)-1 b
Characteristic polynomial (denominator of H(s)): ∆(s)= Det (sI - A)
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order n n poles of the system; depending on matrix A; not depending on chosen
input and output. Denominator order (number of zeroes) less than n (proper systems)
3. General non-linear case
u(t) x& (t )
x(t) y(t)
f(x,u) ∫
x (t ) = x& (t ) dt g(x)
x& (t ) = f(x(t), u(t)) (state equation)
y(t) = g(x(t)) (observation or output equation)
In some cases y(t) = g (x(t); u(t)) → improper system
In simulations, loops of improper systems determine the so called algebraic loops
which are not allowed, since they are undetermined in numerical simulations.
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INTEGRAL: CORE OF A DYNAMIC SYSTEM
Integral is the core element of Ordinary Differential Equations (ODE)
representation and solving.
⋅
u x x
f (x; u)
∫
Integration can be dealt with in many different ways:
dx/dt x(t)
Time domain representation : ∫ dx/dt dt
dx/dt x(t)
Laplace transform representation : 1/s
analogue integrator :
(core of analogue computer)
dx/dt(k ∆t) ∼x(k ∆t)
approximation by a sum : Σ
Z transform representation :
dx/dt(k ∆t) ∼x((k+1)∆t) ∼x(k ∆t)
z-1
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Digital simulation :
(e.g.: accumulator register)
Numerical integration : numerical
(either fixed or varying step) integration
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IMPULSE RESPONSE AND TRANSFER FUNCTION:
LINEAR, TIME INVARIANT SYSTEMS
Time Invariant Systems → parameters are constant
MOVEMENT (time course) is the linear superposition of free movement
and forced movement:
y(t) = yfree(t) + yforced(t)
FREE MOVEMENT: given x(0); with u(t) = 0
yfree(t) = c’eAt x(0)
FORCED MOVEMENT – given u(t) with x(0) = 0:
u(t-τ)
u(t)
U(s)
U(s)
IMPULSE RESPONSE h(t) = c’ eAt b
Laplace transform of impulse response
TRANSFER FUNCTION H(s) = c’ (sI - A)-1 b
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COMPUTATION OF THE FORCED RESPONSE y forced (t) :
- -
Since x(0 ) = 0 , y(0 ) = 0:
- TIME DOMAIN: yforced(t) to an input u(t) is given by the convolution
with the impulse response
- S-TRANSFORMED DOMAIN: Y(s) is given by the product of input
Laplace transform and transfer function = Laplace transform of
impulse response
IF THE SYSTEM IS ASYMPTOTICALLY STABLE
(initial value is forgotten after a “transient”, poles with negative real
part):
- regime behaviour tends to the “particular solution” (it. integrale
particolare), i.e. a solution that satisfies the differential equations
independently from the initial state value
- one and only one particular solution exists for a sinusoidal input:
sinusoidal regime solution;
it is obtained by applying to the input an amplification |H(jω)| and a phase
shift arg(H(jω)); therefore H(jω) is the frequency response =
Fourier transform of impulse response
Transfer function on the imaginary axis, s = jω
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EXAMPLES
Linear model of respiratory mechanics (Fig.2.6)
Linear models of muscle mechanics
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EXAMPLE: HODGKIN-HUXLEY MODEL
THE CIRCUIT DESCRIPTION APPEARS SIMILAR TO THAT OF
PREVIOUS MODELS, BUT:
IS THIS SYSTEM (AT LEAST APPROXIMATELY) LINEAR?
ARE ABOVE RESULTS RELEVANT TO LINEAR TIME INVARIANT
(LTI) SYSTEMS BY ANY MEANS USEFUL?
The main behaviours of the systems (threshold, spiking, refractory period)
relay on the highly non linear dependence of the ionic channel resistances
from membrane polarization.
A linear system could never display these features!
Linearity always requires that parameters are fixed and invariant.
